Optimal. Leaf size=263 \[ -\frac{9 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{8 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{24 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{x \sqrt{\tan ^{-1}(a x)}}{a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \tan ^{-1}(a x)^{3/2}}{3 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{x^3 \sqrt{\tan ^{-1}(a x)}}{6 a c \left (a^2 c x^2+c\right )^{3/2}}-\frac{x^2 \tan ^{-1}(a x)^{3/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}} \]
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Rubi [A] time = 0.64147, antiderivative size = 263, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 10, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.385, Rules used = {4940, 4930, 4905, 4904, 3296, 3305, 3351, 4971, 4970, 3312} \[ -\frac{9 \sqrt{\frac{\pi }{2}} \sqrt{a^2 x^2+1} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{8 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{a^2 x^2+1} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{24 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{x \sqrt{\tan ^{-1}(a x)}}{a^3 c^2 \sqrt{a^2 c x^2+c}}-\frac{2 \tan ^{-1}(a x)^{3/2}}{3 a^4 c^2 \sqrt{a^2 c x^2+c}}+\frac{x^3 \sqrt{\tan ^{-1}(a x)}}{6 a c \left (a^2 c x^2+c\right )^{3/2}}-\frac{x^2 \tan ^{-1}(a x)^{3/2}}{3 a^2 c \left (a^2 c x^2+c\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 4940
Rule 4930
Rule 4905
Rule 4904
Rule 3296
Rule 3305
Rule 3351
Rule 4971
Rule 4970
Rule 3312
Rubi steps
\begin{align*} \int \frac{x^3 \tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^{5/2}} \, dx &=\frac{x^3 \sqrt{\tan ^{-1}(a x)}}{6 a c \left (c+a^2 c x^2\right )^{3/2}}-\frac{x^2 \tan ^{-1}(a x)^{3/2}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{1}{12} \int \frac{x^3}{\left (c+a^2 c x^2\right )^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx+\frac{2 \int \frac{x \tan ^{-1}(a x)^{3/2}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{3 a^2 c}\\ &=\frac{x^3 \sqrt{\tan ^{-1}(a x)}}{6 a c \left (c+a^2 c x^2\right )^{3/2}}-\frac{x^2 \tan ^{-1}(a x)^{3/2}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{2 \tan ^{-1}(a x)^{3/2}}{3 a^4 c^2 \sqrt{c+a^2 c x^2}}+\frac{\int \frac{\sqrt{\tan ^{-1}(a x)}}{\left (c+a^2 c x^2\right )^{3/2}} \, dx}{a^3 c}-\frac{\sqrt{1+a^2 x^2} \int \frac{x^3}{\left (1+a^2 x^2\right )^{5/2} \sqrt{\tan ^{-1}(a x)}} \, dx}{12 c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{x^3 \sqrt{\tan ^{-1}(a x)}}{6 a c \left (c+a^2 c x^2\right )^{3/2}}-\frac{x^2 \tan ^{-1}(a x)^{3/2}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{2 \tan ^{-1}(a x)^{3/2}}{3 a^4 c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \frac{\sin ^3(x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{12 a^4 c^2 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \int \frac{\sqrt{\tan ^{-1}(a x)}}{\left (1+a^2 x^2\right )^{3/2}} \, dx}{a^3 c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{x^3 \sqrt{\tan ^{-1}(a x)}}{6 a c \left (c+a^2 c x^2\right )^{3/2}}-\frac{x^2 \tan ^{-1}(a x)^{3/2}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{2 \tan ^{-1}(a x)^{3/2}}{3 a^4 c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \left (\frac{3 \sin (x)}{4 \sqrt{x}}-\frac{\sin (3 x)}{4 \sqrt{x}}\right ) \, dx,x,\tan ^{-1}(a x)\right )}{12 a^4 c^2 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \sqrt{x} \cos (x) \, dx,x,\tan ^{-1}(a x)\right )}{a^4 c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{x^3 \sqrt{\tan ^{-1}(a x)}}{6 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{x \sqrt{\tan ^{-1}(a x)}}{a^3 c^2 \sqrt{c+a^2 c x^2}}-\frac{x^2 \tan ^{-1}(a x)^{3/2}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{2 \tan ^{-1}(a x)^{3/2}}{3 a^4 c^2 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \frac{\sin (3 x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{48 a^4 c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{16 a^4 c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \frac{\sin (x)}{\sqrt{x}} \, dx,x,\tan ^{-1}(a x)\right )}{2 a^4 c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{x^3 \sqrt{\tan ^{-1}(a x)}}{6 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{x \sqrt{\tan ^{-1}(a x)}}{a^3 c^2 \sqrt{c+a^2 c x^2}}-\frac{x^2 \tan ^{-1}(a x)^{3/2}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{2 \tan ^{-1}(a x)^{3/2}}{3 a^4 c^2 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \sin \left (3 x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{24 a^4 c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{8 a^4 c^2 \sqrt{c+a^2 c x^2}}-\frac{\sqrt{1+a^2 x^2} \operatorname{Subst}\left (\int \sin \left (x^2\right ) \, dx,x,\sqrt{\tan ^{-1}(a x)}\right )}{a^4 c^2 \sqrt{c+a^2 c x^2}}\\ &=\frac{x^3 \sqrt{\tan ^{-1}(a x)}}{6 a c \left (c+a^2 c x^2\right )^{3/2}}+\frac{x \sqrt{\tan ^{-1}(a x)}}{a^3 c^2 \sqrt{c+a^2 c x^2}}-\frac{x^2 \tan ^{-1}(a x)^{3/2}}{3 a^2 c \left (c+a^2 c x^2\right )^{3/2}}-\frac{2 \tan ^{-1}(a x)^{3/2}}{3 a^4 c^2 \sqrt{c+a^2 c x^2}}-\frac{9 \sqrt{\frac{\pi }{2}} \sqrt{1+a^2 x^2} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{8 a^4 c^2 \sqrt{c+a^2 c x^2}}+\frac{\sqrt{\frac{\pi }{6}} \sqrt{1+a^2 x^2} S\left (\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )}{24 a^4 c^2 \sqrt{c+a^2 c x^2}}\\ \end{align*}
Mathematica [C] time = 1.05531, size = 272, normalized size = 1.03 \[ \frac{3 \left (a^2 x^2+1\right )^{3/2} \left (3 \sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-i \tan ^{-1}(a x)\right )+3 \sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},i \tan ^{-1}(a x)\right )+\sqrt{3} \left (\sqrt{-i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},-3 i \tan ^{-1}(a x)\right )+\sqrt{i \tan ^{-1}(a x)} \text{Gamma}\left (\frac{1}{2},3 i \tan ^{-1}(a x)\right )\right )\right )-7 \sqrt{6 \pi } \left (a^2 x^2+1\right )^{3/2} \sqrt{\tan ^{-1}(a x)} \left (3 \sqrt{3} S\left (\sqrt{\frac{2}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )-S\left (\sqrt{\frac{6}{\pi }} \sqrt{\tan ^{-1}(a x)}\right )\right )+24 \tan ^{-1}(a x) \left (a x \left (7 a^2 x^2+6\right )-2 \left (3 a^2 x^2+2\right ) \tan ^{-1}(a x)\right )}{144 a^4 c \left (a^2 c x^2+c\right )^{3/2} \sqrt{\tan ^{-1}(a x)}} \]
Warning: Unable to verify antiderivative.
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Maple [F] time = 3.464, size = 0, normalized size = 0. \begin{align*} \int{{x}^{3} \left ( \arctan \left ( ax \right ) \right ) ^{{\frac{3}{2}}} \left ({a}^{2}c{x}^{2}+c \right ) ^{-{\frac{5}{2}}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: RuntimeError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{3} \arctan \left (a x\right )^{\frac{3}{2}}}{{\left (a^{2} c x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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